Analysis and Probability Seminar

The seminar is joint for the division of Analysis and Probability and its main themes are Mathematical Physics, Probability Theory and Harmonic and Functional Analysis. The seminar is encouraged to be aimed at a broader audience, but it may be of a more specialised nature when indicated in the last line of the abstract. The talks in the seminar are usually 60 minutes, but can also be in the format 2x45 minutes when indicated in the abstract.

The seminar takes place Tuesdays at 1315-1415 in MV:L15 unless otherwise indicated.

Should you have any questions or suggestions, feel free to email one of the organizers Martin Hallnäs (Mathematical Physics), Erik Broman (Probability Theory) or Magnus Goffeng (Harmonic and Functional Analysis).

2019-12-10 (13.15-15.00)
Erik Duse (KTH)

On the geometry and regularity of frozen boundaries in dimer models
In this talk I will discuss a problem in the calculus of variations arising from dimer models. These are two dimensional statistical mechanical models that can be described in terms of random Lipschitz function. In the case when the systems become infinitely large, these random functions converge almost surely to a deterministic function, called the asymptotic height function. This function can be characterised as the unique minimiser of convex functional together with a natural convex gradient constraint.
These functionals arising from dimer models are not differentiable and therefore are very chal- lenging to study. In particular the associated Euler-Lagrange equation are non-linear elliptic degenerate and singular partial differential equation.
A very interesting feature of dimer models is that they give rise to so called frozen bound- aries. For example, a question one would like to answer is the regularity and geometry of these boundaries as well as the regularity of the minimisers. Using a reduction of the Euler-Lagrange equations to first order systems of Beltrami equations I will show in this talk how the degeneracy and non-linearity of the equations can be tamed and how one can give an essentially complete characterisation of the qualitative properties of the minimisers.
This is based on joint work with Kari Astala, István Prause and Xiao Zhong.

Prerequisites: Basic notions from convex analysis, e.g., subdifferentiability etc. Basic knowledge of elliptic partial differential equations and calculus of variations. Solid grounding in complex analysis, e.g., Riemann mapping theorem, Koebe uniformization, Cauchy-Riemann equations, Hardy spaces. Some knowledge of quasi-conformal mappings is good but not essential.

Previous seminars

2019-01-15
Ramiz Reza

A class of Hausdorff moment sequences and its application
We consider a class of sequences, namely q(n)/p(n), where q and p are polynomial with real coefficient and degree of q is less than degree of p. We try to investigate when such sequence is a Hausdorff moment sequence. We provide a necessary condition ( involving zeros of p ) for 1/p(n) to be a Hausdorff moment sequence. Using this necessary condition we show that Schur product of Szego kernel on unit disc with a subnormal reproducing kernel need not be a subnormal kernel. Consider the class of all real polynomial p of degree 5 such that all of its root has same real part. We provide a complete description of all polynomial p in this class so that 1/p(n) becomes a Hausdorff moment sequence.

The talk is aimed at a general analysis/probability audience.

2019-01-22 in MV:L15
Przemysław Ohrysko

Spectral radius formula for Fourier-Stieltjes algebras
In my talk I will present a short proof of the spectral radiu formula for a measure algebra on a locally compact Abelian group which can be easily generalized to Fourier-Stieltjes algebras of non-commutative locally compact groups. The second part will be devoted to proving that there are no non-trivial relations between the supremum of the Fourier-Stieltjes transform, the spectral radius and the norm of the measure. The talk is based on a joint work with Maria Roginskaya (preprint is available on arxiv.org with the identifier: 1811.11560).

Basic knowledge in Banach algebras and commutative harmonic analysis is welcome but the major part of the talk will be completely elementary and so accessible to everyone.

2019-01-29 in MV:L15
Peter Sjögren

Sharp estimates of the spherical heat kernel
We prove sharp two-sided global estimates for the heat kernel associated with a Euclidean sphere of arbitrary dimension. The proof relies on the classical ultraspherical polynomials and the corresponding heat kernel. This is joint work with A. Nowak and T. Z. Szarek. The seminar will be short, at most 45 minutes.

It is comprehensible with only very basic knowledge about the heat equation. It will be helpful but not necessary to have heard about ultraspherical (also called Gegenbauer) polynomials.

2019-02-05
Mateusz Wasilewski (Katholieke Universiteit Leuven)

Quantum isomorphisms of graphs and monoidal equivalence
I will talk about a graph isomorphism game and how different frameworks of quantum strategies lead to several notions of a quantum isomorphism between graphs; these will all be encompassed by a certain*-algebra associated to the game. Next I will introduce the quantum automorphism group of a graph and define an equivalence between these objects called the monoidal equivalence. Then I will present a connection between the *-algebra associated to the game and a *-algebra associated to monoidal equivalence, called the linking algebra, and how it allows to prove that a very weak form of quantum isomorphism automatically implies a much stronger version thereof. If time permits, I plan to discuss some open problems. This is joint work with Brannan, Chirvasitu, Eifler, Harris, Paulsen, and Su.

The talk will be aimed at an audience familiar with basic operator theory. Some background in operator algebras will be helpful but not necessary.

2019-02-19
Hjalmar Rosengren

Elliptic lattice models
In statistical mechanics, there are many solvable or integrable lattice models with intriguing mathematical properties. Typically, the most general cases of these models are described by elliptic functions. I will give an introduction to this topic, focusing on the fundamental examples of the eight-vertex model and the XYZ spin chain. Most of the talk will review classical work of Baxter and Sutherland from the 1970s. If time permits, I will briefly discuss ongoing joint work with Christian Hagendorf on nearest neighbour correlations of the supersymmetric XYZ chain.

The talk is aimed at a general mathematical audience.

2019-03-05
Stephan Wagner (Stellenbosch University)

Loop models on a fractal
We consider two types of loop models on self-similar, fractal-like graphs, the Sierpiński gasket and its finite approximations being a classical example. In one model, a 2-factor (spanning subgraph whose components are cycles) is chosen uniformly at random. In the other, the edge set is partitioned into cycles, again uniformly at random. The presence of "holes" in the graph turns out to have interesting consequences. While the latter model mostly yields rather short cycles, long cycles surrounding the holes appear with high probability in random 2-factors, and those long loops feature interesting geometric properties reminiscent of random walks and their loop-erased variant. Moreover, an interesting phase transition can be observed.

This talk is aimed at the whole division.

2019-04-02
Yilin Wang (ETH Zürich)

Loewner energy, determinants of Laplacians and Brownian loop measure
We have introduced Loewner energy to describe the large deviation behaviour of chordal Schramm-Loewner evolution with vanishing parameter. The parametrization-independence of its generalization to Jordan curves was explained by an intrinsic description using zeta-regularized determinants of Laplacians which can be interpreted as the mass of Brownian loop measure. This identity suggests that one may interpret the Loewner energy as the total mass of the Brownian loops attached to the curve. I will show that the cut-off by equipotentials provides an appropriate renormalization to make sense of this statement.

The talk is aimed at the whole division.

2019-04-09
Lachlan MacDonald (University of Wollongong)

Dynamical invariants of foliated manifolds
A foliation of a manifold M specifies the set of integral submanifolds (referred to in foliation theory as "leaves") to some nonsingular system of differential equations on M. The "holonomy" of integral curves in a foliated manifold gives rise to an intrinsic dynamical system, whose topology has implications for the global behaviour of solutions to the corresponding system of DEs. In this talk, I will give a primer on foliations and their associated dynamical invariants, as well as discuss how this information can be accessed in the spirit of the Atiyah-Singer index theorem.

The talk is aimed at an audience familiar with the basics of manifolds and vector bundles. Some knowledge of algebraic topology and operator algebras will be helpful but not necessary.

2019-05-07
Michael Björklund

Fun with i.ni.d. random variables
Given a countable group G and a function f : G \ra (0,1) we consider a sequence (x_g) of independent {0,1} random variables, indexed by g \in G, where P(x_g = 0) = f(g). If f is constant, or “close to” a constant, then this sequence is “essentially” stationary. If f is varying very rapidly, then no “stationary” behaviour can be seen in the sequence. However, on a large class of groups (namely those with non-trivial first L^2-cohomology), there are interesting choices of f, which are far from constant, but for which some “stationary” behaviour is retained. We shall discuss this class of groups and the various f’s that give rise to interesting “almost stationary”, but not stationary, processes.

The talk is intended for a general audience, and is based on joint works with Zemer Kosloff (Jerusalem) and Stefaan Vaes (Leuven).

2019-05-16 (Thursday) at 10:00 in MV:L15
Andreas Rosén

Maxwell scattering with the help of Dirac
This talk concerns numerical computation of the electromagnetic field scattered by in general non-smooth objects, through the methods of integral equations on the boundary. There are three main steps: 1. Integral formulation. 2. Discretization 3. Fast solvability of dense linear systems. We are primarily concerned with 1, and report on work in progress joint with Johan Helsing (Lund), using his state-of-the-art algorithms for 2. The new integral formulations
we are testing are obtained using Clifford algebra and hypercomplex Hardy spaces. In this first test we have considered two-dimensional TM scattering, and in particular the computation of surface plasmon excitations by corner singularities. For the integral equation this means that we hit the essential spectrum.

This talk is intended for a general analysis/numerical analysis audience.

2019-05-20 (Monday) at 13:15
Nikoletta Louca (Heriot-Watt University, Edinburgh)

The magnitude function on domains
The notion of magnitude is a real-valued invariant of metric spaces that has been introduced by Leinster, and is analogous to the Euler characteristic of topological spaces and the cardinality of sets. Basic questions about the geometric properties of the magnitude function still remain open, even for compact subsets of R^n. In this talk we discuss some of the properties of the magnitude function on smooth domains of (mainly odd-dimensional) manifolds, part of which is work in progress with Heiko Gimperlein and Magnus Goffeng.

The talk is aimed at a general analysis audience. Basic knowledge of pseudodifferential operators will be useful but not assumed.

2019-05-21
Przemysław Ohrysko

Spectrally reasonable measures II
Let G be a locally compact Abelian group. A complex measure on G is said to have a natural spectrum if its spectrum (in a standard Banach algebras sense) is equal to the closure of the range of its Fourier-Stieltjes transform. It is well-known that on every locally compact non-discrete Abelian group there exists a measure with non-natural spectrum (Wiener-Pitt phenomenon). The structure of the set of all measures with a natural spectrum was studied by Hatori and Sato who proved that this set span the whole algebra of measures and hence cannot be closed under addition. Inspired by this result, the notion of spectrally reasonable measure was introduced as a measure which perturbs any measure with a natural spectrum to a measure with a natural spectrum. It turned out that the set of all spectrally reasonable measures has Banach algebra structure.

In my talk, I will present the recent developments in this area: the complete characterization of spectrally reasonable measures for classical groups of harmonic analysis – the Euclidean spaces and the torus. The first of the presentation will be devoted to the general discussion and advertisement of the results and in the second part I will show the details of the proof of the classification of spectrally reasonable measures for Euclidean spaces. The arguments are quite involved and use the tools from classical harmonic analysis, Banach algebra theory and some nearly completely forgotten results from the general topology.

The talk is based on the preprint with the same title, available on arxiv.org with the identifier: 1903.04853 written in collaboration with Michał Wojciechowski.

This seminar meeting is aimed at everybody familiar with basics of Banach algebra and classical harmonic analysis.

2019-05-28
Ksenia Fedosova (University of Freiburg)

Eisenstein series and Selberg zeta function twisted by representations with non-expanding cusp monodromy
In this talk, we investigate the behaviour of the Eisenstein series, or generalized eigenfunctions of the Laplace operator on hyperbolicsurfaces. We twist them by a (possibly) non-unitary representation of the fundamental group of the manifold, show their convergence on some half-plane and study their Fourier expansion. Further, we show the convergence and meromorphic continuation of the Selberg zeta function twisted by the same type of representations.

The talk is aimed at a general analysis audience.

2019-06-05 (Wednesday) at 10:00
Jörg Weber (University of Bayreuth)

The Relativistic Vlasov-Maxwell System with Boundary Conditions and External Currents
The time evolution of a collisionless plasma is modelled by the relativistic Vlasov-Maxwell system, which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We consider the case that the plasma particles are located in a bounded domain, for example a fusion reactor. In the exterior, there are external currents that may serve as a control of the plasma if adjusted suitably. Also, we allow material parameters, namely permittivity and permeability, which may depend on the space coordinate.

In the first part of the talk, we discuss the solution theory of the modeling nonlinear PDE system and prove existence of global-in-time weak solutions.

In the second part, we consider a minimizing problem: A typical aim is to drive the amount of particles hitting the boundary of their container and thus causing damage to a minimum, while the control costs should be kept as small as possible. We prove existence of minimisers of the arising minimizing problem and give an approach to derive first order optimality conditions.

The talk is aimed at an audience familiar with mathematical physics.

2019-06-11
Roger Behrend (Cardiff University)

The combinatorics of alternating sign matrices
An alternating sign matrix (ASM) is a square matrix in which each entry is -1, 0 or 1, and along each row and column the nonzero entries alternate in sign, starting and ending with a 1. It was conjectured by Mills, Robbins and Rumsey in 1982 that the number of ASMs of fixed size is given by a certain simple product formula. A relatively short proof of this conjecture was obtained by Kuperberg in 1996, using connections with the six-vertex model of statistical mechanics. It was also conjectured in the 1980s that the number of ASMs of fixed odd size which are invariant under diagonal and antidiagonal reflection is given by a simple product formula. In this talk, I will provide an introduction to ASMs, a review of Kuperberg's proof of the ASM enumeration formula, and an outline of a recent proof of the enumeration formula for diagonally and antidiagonally symmetric ASMs of odd size.

The talk is aimed at the whole division.

2019-08-27
Antar Bandyopadhyay (Indian Statistical Institute (Delhi Centre))

Random Recursive Tree, Branching Markov Chains and Urn Models
In this talk, we will show that a branching Markov chain defined on the random recursive tree is nothing but a balanced urn model. This is a novel connection between these two apparently unrelated probabilistic models. Exploring the connection further we will derive fairly general scaling limits for balanced urn models with colors indexed by any Polish Space. Thus generalizing the classical urn models defined on finitely many colors to a measure valued process on a general Polish Space. We will use the connection to show that several exiting results on classical finite color urn schemes, as well as, recently introduced infinite color urn schemes, can be derived easily from the general asymptotic. In particular, we will prove that the well know classical result of the finitely many color balanced urn, when the replacement matrix is irreducible will also holds for countably infinite many colors (under certain technical assumptions). We will also discuss the special case, of null recurrent replacement matrices, which can only arise in the context of infinitely many colors. We will further show that the connection can also be used to derive exact asymptotic of the sizes of the connected components of a random recursive forest, which is obtained by removing the root of a random recursive tree.

(Parts are joint work with Debleena Thacker and Svante Janson)

The talk is aimed at a general probability audience.

2019-09-10
Joakim Arnlind (Linköpings Universitet)

Pseudo-Riemannian calculi and noncommutative connections
In a series of papers, we have developed Pseudo-Riemannian calculi as a framework for discussing Levi-Civita connections and curvature in noncommutative geometry. One of the original motivations was to understand under what conditions a torsion-free and metric connection is unique, and to what extent the curvature tensor exhibits the classical symmetries. Moreover, it turns out that one may develop a theory of submanfolds and minimal embeddings in the spirit of differential geometry, finding noncommutative analogues of the second fundamental form, Gauss' equations, etc. In this talk, I will give an overview of Psuedo-Riemannian calculi together with a few concrete examples. This is joint work with Mitsuru Wilson and Axel Tiger Norkvist.

The talk is aimed at a general audience, but a basic knowledge of algebra and differential geometry will be helpful.

2019-09-17
Yanqi Qiu (Chinese Academy of Sciences)

A class of positive functions on the circle and branching-type stationary stochastic processes
We propose a definition of branching-type stationary stochastic processes on rooted trees. The spectral measure of such stochastic processes are related to positive definite functions on positive integers. After giving a complete criterion of such positive definite functions in the case of rooted homogeneous trees, we are going to talk about their applications on inequalities of Hankel operators. The talk is based on joint work with Zipeng Wang.

This talk is aimed for probability and analysis audience.

2019-09-24 at 10am in MV:L14
Joint AP/KASS Seminar:
Richard Schoen (University of California, Irvine)

Positive scalar curvature, the Dirac operator, and minimal hypersurfaces
We will discuss the approaches which have been successful in understanding manifolds with positive scalar curvature. Two basic and related questions one can ask are the topological classification of such manifolds and the positive mass theorem. The two approaches which have worked in all dimensions are the Dirac operator and the minimal hypersurface approach. Both have many successes but also limitations which we will describe. In the case of three dimensions there are also the inverse mean curvature flow and the Ricci flow which yield strong results.

2019-09-26 Thursday at 15:15 in MV:L15
Joint AP/AGNT Seminar:
Wee Teck Gan (National University of Singapore)

Triality and functoriality
I will discuss some implications of the existence of the triality automorphism in the Langlands program, such as the demonstration of certain instances of functorial lifting. This is joint work with Gaetan Chenevier.

2019-10-08 Tuesday at 1515
Andreas Juhl (Humboldt-Universität Berlin)

Branson-Gover operators and residue families.
In 2004, T. Branson and R. Gover introduced differential operators acting on differential forms which generalize the conformal powers of the Laplacian (GJMS operators). Their construction rests on tractor calculus. We found that these constructions admit an alternative description in terms of a generalization of the notion of residue families. In the scalar case, this notion has been introduced by the author in 2009. In the form case, residue families are conformally covariant one-parameter families of differential operators which are curved versions of so-called symmetry breaking operators (a concept recently introduced by T. Kobayashi). In the seminar, we describe these concepts, their interactions and some of the emerging new results.

2019-10-15
Joao Pedro Paulos (Chalmers/GU)

Classical Descriptive Set Theory and Closed Sets of Uniqueness
Descriptive Set Theory (DST) is the study of Polish spaces and its subsets. Even though its ethos is indissociable from questions coming from Set Theory and Logic, the development of DST lead to many applications within a wide range of topics emerging from Harmonic Analysis, Recursion Theory, Functional Analysis or Dynamical Systems. In this talk, we introduce some elementary concepts and prove a result on the complexity of the set of closed sets of uniqueness

The talk is aimed at a general analysis/probability audience.

2019-10-22
Antti Perälä (Chalmers/GU)

Integration operators from Bergman spaces to Hardy spaces of the unit ball
We completely characterize the boundedness of the Volterra type integration operators acting from the weighted Bergman spaces to the Hardy spaces of the unit ball. A partial solution to this problem in the one-dimensional setting was previously obtained by Zhijian Wu. We solve the missing cases and extend the results to all dimensions. Our tools involve area methods from harmonic analysis, Carleson measures and Kahane-Khinchine type inequalities, as well as techniques and integral estimates related to Hardy and Bergman spaces. This talk is based on a joint paper with Santeri Miihkinen, Jordi Pau and Maofa Wang.

The talk is aimed at an audience familiar with holomorphic function spaces and operator theory. (The key concepts will be explained during the talk.)

2019-10-29
Nikoletta Louca and Heiko Gimperlein (Heriott-Watt University, Edinburgh)

Boundary problems for the fractional Laplacian: a new approach
Solutions to elliptic boundary value problems on polygonal domains exhibit singularities at edges and corners. On the other hand, there has been much recent interest in nonlocal equations, e.g. involving the fractional Laplacian. We discuss a unified approach to their analysis, based on pseudodifferential techniques developed for elliptic differential equations with mixed boundary conditions. For the fractional Dirichlet problem in a domain with smooth boundary, our approach recovers recent results of G. Grubb, using independent methods. In addition to the analysis, applications of the results to robotics and the design of efficient numerical methods will be mentioned. (Joint with Rafe Mazzeo and Jakub Stocek.)

The talk should be accessible and of interest to a general analysis audience. It will also discuss applied and numerical implications.GothenburgOct19.pdf[Slides]

2019-11-07 (Obs: Torsdag)
Jakob Björnberg (Chalmers/GU)
The interchange process with reversals
The interchange process is a model for random permutations formed by composing random transpositions. Here we consider a variant of the interchange process where a fraction of the transpositions are replaced by `reversing transpositions'. A motivation for studying such processes is that they appear in the study of quantum models for magnetism, but they are also interesting in their own right. We will discuss a recent result obtained together with M. Kotowski, B. Lees and P. Milos which describes the scaling limit of the joint distribution of the largest cycles for the process defined on the complete graph.

This talk is directed to those with a general knowledge of probability theory

​​​​​2019-11-12
Bernt Wennberg (Chalmers/GU)

A Kac model with an exclusion principle
The Boltzmann equation is one of the fundamental equations in non-equilibrium statistical mechanics, and still after almost 150 years a great challenge for mathematicians: there is still a lot to do concerning existence and uniqueness of solutions, for example, and a satisfactory, rigorous, derivation of the equation from a particle system is still lacking. To better understand the question of derivation of the equation, Marc Kac invented a toy model of an N-particle system, and carried out a complete proof of validation of a related Boltzmann equation for this model. In my talk I will give a very basic introduction to the Boltzmann equation, and the validation problem. I will describe the Kac model, and I will present a modification with an exclusion principle, and analyse the equilibrium configurations for this model. The last part is a collaboration with Eric Carlen, Rutgers.

The talk is aimed at a general audience with an interest in analysis and probability theory, and requires now previous acquaintance with the Boltzmann equation.

2019-11-19
Alexander Gorokhovsky (University of Colorado, Boulder)

Cyclic cohomology and higher indices
We show that a careful analysis of cyclic cohomology of the algebra of pseudodifferential operators on a closed manifold allows one to better understand and generalize classical Helton-Howe theorems as well as obtain new invariants of pseudodifferential operators. This is a joint work with H. Moscovici.

2019-11-21, Thursday 1315 in MVF23
Alexey Kuzmin (Chalmers/GU)

Classification of C* algebras of deformed canonical anticommutation relations
In my talk I consider deformation of the CAR C*-algebra by a skew-symmetric matrix \Theta. This C*-algebra appears to have a structure of a noncommutative bundle over a hypercube with fibers essentially isomorphic to noncommutative tori. This geometric structure is used in order to classify representation theory and to show that two deformed CAR C*-algebras are isomorphic if and only if their deformation matrices \Theta_1, \Theta_2 are in the same orbit of the action of the symmetric group on the space of skew-symmetric matrices. The talk is based on a joint work with Lyudmila Turowska. 

The talk will be aimed at an audience familiar with basic theory of C*-algebras.

2019-11-26 at 15:15
Håkan Andreasson (Chalmers/GU)

On the existence and structure of stationary solutions of the Einstein-Vlasov system.
The present status on the existence and structure of static and stationary solutions of the Einstein-Vlasov system is reviewed. The structure of spherically symmetric static solutions is quite well understood. I will discuss a number of their features, in particular the case of charged solutions since they indicate what properties can be expected of axisymmetric stationary solutions. Existence of axially symmetric stationary solutions that are perturbed off from spherically symmetric Newtonian solutions have been obtained analytically whereas solutions far from being spherically symmetric have only been constructed numerically. I will discuss the properties of the latter. In particular, two different sequences of toroidal solutions which contain ergoregions will be described in detail. These solutions either approach an extreme Kerr black hole or they have the property that the geometry becomes conical in the limit and such solutions may provide models of cosmic strings.

The talk will be aimed at a general analysis audience.

2019-12-03 (13.15-15.00)
Grigori Rozenblioum (Chalmers/GU)

Spectral properties of the Neumann-Poincare operator in Electrostatics and Elastostatics and their relation to metamaterials
The Neumann-Poincare operator is an integral operator arising in the analysis of boundary problems for elliptic equations. It was proposed in the end of 19th century by C.Neumann and H.Poincare. The interest to the spectral properties of this operator was renewed recently due to its relation to the so-called metamaterials, artificial composite materials with wrong sign of their physical characteristics. In the talk we will present some information about these metamaterials and their relation to NP operators as well some recent results about spectral properties (especially, eigenvalues behavior) of these operators.

Published: Mon 12 Dec 2016. Modified: Mon 09 Dec 2019